Explore and calculate values based on fundamental geometric properties of circles.
Last updated: 2026-05-24T22:58:31.600Z
The angle subtended by an arc at the center is twice the angle subtended by it at any point on the remaining part of the circle.
Enter an angle in degrees (numeric). Inputs must be valid numbers.
Center Angle
60°
Circle theorems are geometric rules that describe relationships between angles, arcs, chords, and tangents within a circle. These theorems are fundamental to understanding circle geometry and have applications in architecture, engineering, and navigation.
By mastering circle theorems, you can calculate unknown angles and distances without direct measurement, relying instead on the inherent symmetry and properties of circles.
A straight line segment whose endpoints both lie on the circumference. The diameter is the longest chord.
A straight line that touches the circle at exactly one point without entering the interior.
A portion of the circumference of a circle between two points on the circle.
A region of a circle bounded by a chord and the arc subtended by the chord.
Find the angle at the center when the angle at the circumference is 35°:
Given: Circumference angle = 35°
Step 1: Apply the theorem: Center Angle = 2 × Circumference Angle
Step 2: Substitute: Center Angle = 2 × 35°
Step 3: Calculate: Center Angle = 70°
Result:
70°
A chord is a straight line segment whose endpoints both lie on the circumference of a circle. The diameter is the longest possible chord.
A tangent is a straight line that touches the circle at exactly one point and never enters the interior of the circle.
A cyclic quadrilateral is a four-sided polygon whose vertices all lie on a single circle. Its opposite angles always sum to 180°.
No. These theorems are specific to circles. Ellipses have different geometric properties and do not follow these same rules.
An inscribed angle is half the central angle that subtends the same arc. This is the foundation of the angle-at-center theorem.
Circle symmetry is fundamental to these theorems. Circles have rotational symmetry about their center, which creates equal relationships.
Yes. Many circle theorems can be used in reverse to construct circles through given points or with specific angle properties.
Absolutely. They're used in architecture (domes, arches), engineering (bearings, gears), navigation, and computer graphics.